🧮 零空间（核）计算器Null Space Calculator
用分数精确计算矩阵的零空间（核）：化为行最简形（RREF），找出主元列和自由变量，给出零空间的一组基向量（可化为整数），以及秩和零度。Find the null space (kernel) of a matrix exactly with fractions: reduce it to row echelon form (RREF), identify pivot columns and free variables, and get a basis for the null space (optionally scaled to integers) with the rank and nullity.
使用说明与常见问题How-to & FAQ
「零空间（核）计算器」是什么？What is the Null Space Calculator tool?
用分数精确计算矩阵的零空间（核）：化为行最简形（RREF），找出主元列和自由变量，给出零空间的一组基向量（可化为整数），以及秩和零度。Find the null space (kernel) of a matrix exactly with fractions: reduce it to row echelon form (RREF), identify pivot columns and free variables, and get a basis for the null space (optionally scaled to integers) with the rank and nullity.
如何使用「零空间（核）计算器」？How do I use the Null Space Calculator tool?
在「零空间（核）计算器」中，填写上方字段等参数，结果会随输入实时计算并显示在下方。In the Null Space Calculator: Fill in the fields above; results are recalculated instantly as you type.
零空间怎么求？How is the null space found?
解齐次方程组 Ax = 0：把 A 化成行最简形，主元列对应的变量用自由变量表示；每个自由变量取 1、其余自由变量取 0，就得到一个基向量。基向量的个数（零度）= 列数 − 秩。Solve Ax = 0: reduce A to RREF and write the pivot variables in terms of the free ones. Setting one free variable to 1 and the others to 0 gives each basis vector; their number (the nullity) = columns − rank.
为什么结果是分数或整数？Why fractions or integers?
全程用分数精确计算，没有浮点误差。勾选「化为整数」时，每个基向量乘以分母的最小公倍数再约去公因数；这不改变它张成的空间。Everything is exact rational arithmetic, with no floating-point error. With integer scaling, each basis vector is multiplied by the LCM of its denominators and divided by the common factor — the span is unchanged.
「零空间（核）计算器」免费吗？需要注册吗？Is the Null Space Calculator free? Do I need an account?
「零空间（核）计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Null Space Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.